Proving that all real numbers lie in Fatou set of $f(z)=\sin(z)/3$

Question

The problem is this:

Define the entire function $f$ by $$f(z)=\frac{\text{sin}(z)}{3}.$$ Show that all real numbers lie in the Fatou set $F_f$.

Hint: For $x\in \mathbb{R}$ we have $|\text{sin}(x)|\leq |x|$.

I have managed to show that all multiples of $\pi$ are in the Fatou set using complete invariance, since 0 is a fixpoint and these all map to 0. Also that the iterates tend to 0 for all real x, using the hint. But this does not immediately imply that they are in the Fatou set, does it?

Beyond that, I am a bit stuck. Any help would be greatly appreciated.

Answer 1

First, $|f'(0)| = \frac 13 < 1$ so that $z=0$ is an attracting fixed point of $f$, and some disk $B_r(0)$ is contained in the Fatou set $F(f)$.

Then, as you noticed, for every $x \in \Bbb R$ the iterates $f^{n}(x)$ converge to zero for $n \to \infty$. It follows that $f^{n}(x) \in B_r(0) \subset F(f)$ for some $n$, and therefore $x \in F(f)$ because of the invariance of the Fatou set.